Abstract

We implement a Langevin approach for the transport of charm quarks in
the UrQMD (hydrodynamics + Boltzmann) hybrid model. Due to the
inclusion of event-by-event fluctuations and a full (3+1) dimensional
hydrodynamic evolution, this approach provides a more realistic model
for the evolution of the matter produced in heavy ion collisions as
compared to simple homogeneous fireball expansions usually employed.
As drag and diffusion coefficients we use a resonance approach for
elastic heavy-quark scattering and assume a decoupling temperature of
the charm quarks from the hot medium of 130MeV. A coalescence
approach at the decoupling temperature for the hadronization of the
charm quarks to D-mesons is also included. We present calculations of
the nuclear modification factor RAA as well as the elliptic flow
v2 in Pb+Pb collisions at √sNN=2.76TeV. The
comparison to ALICE measurements shows a very good agreement with our
calculations.

One major goal of high-energy heavy-ion physics is to recreate the phase
of deconfined matter, where quarks and gluons more quasi free (the Quark
Gluon Plasma, QGP) as it might have existed a few microseconds after the
Big Bang. Various experimental facilities have been built to explore the
properties of this QGP experimentally, while on the theory side a
multitude of (potential) signatures and properties of the QGP have been
predicted Adams et al. (); Adcox et al. (); Müller et al. (2012).

Heavy quarks are an ideal probe for the QGP. They are produced in the
primordial hard collisions of the nuclear reaction and therefore probe
the created medium during its entire evolution process. When the system
cools down they hadronize, and their decay products can finally be
detected. Therefore, heavy-quark observables provide new insights into
the interaction processes within the hot and dense medium. Two of the
most interesting observables are the elliptic flow, v2, and the
nuclear modification factor, RAA, of open-heavy-flavor mesons and
their decay products like “non-photonic” single electrons. The
measured large elliptic flow, v2, of open-heavy-flavor mesons and the
“non-photonic single electrons or muons” from their decay underline
that heavy quarks take part in the collective motion of the bulk medium,
consisting of light quarks and gluons. The nuclear modification factor
shows a large suppression of the open-heavy flavor particles’ spectra at
high transverse momenta (pT) compared to the findings in pp
collisions. This also supports a high degree of thermalization of the
heavy quarks with the bulk medium. A quantitative analysis of the degree
of thermalization of heavy-quark degrees of freedom in terms of the
underlying microscopic scattering processes thus leads to an
understanding of the mechanisms underlying the large coupling strength
of the QGP and the corresponding transport properties.

The UrQMD hybrid model has been developed to combine the advantages of
transport theory and (ideal) fluid dynamics Petersen et al. (2008). It
uses initial conditions, generated by the UrQMD model
Bass et al. (1999); Dumitru et al. (1999), for a full (3+1) dimensional ideal
fluid dynamical evolution, including the explicit propagation of the
baryon current. After a Cooper-Frye transition back to the transport
description, the freeze out of the system is treated dynamically within
the UrQMD approach. The hybrid model has been successfully applied to
describe particle yields and transverse dynamics from AGS to LHC
energies
Petersen et al. (2008); Steinheimer et al. (2008, 2010); Petersen et al. (2010); Petersen (2011)
and is therefore a reliable model for the flowing background medium.

The equation of state we use for our calculations includes quark and
gluonic degrees of freedom coupled to a hadronic parity-doublet model
Steinheimer et al. (2011). It includes a smooth crossover at low baryon
densities between an interacting hadronic system and a quark gluon
plasma. The thermal properties of the EoS are in agreement with lattice
QCD results at vanishing baryon density, and the EoS therefore is well
suited for our investigation at LHC energies.

The diffusion of a “heavy particle” in a medium consisting of “light
particles” can be described with help of a Fokker-Planck equation
Svetitsky (1988); Mustafa et al. (1998); Moore and Teaney (2005); van Hees and Rapp (2005); van Hees et al. (2006, 2008); Gossiaux and Aichelin (2008); He et al. (2011)
as an approximation of the collision term of the corresponding Boltzmann
equation. It can be mapped into an equivalent stochastic Langevin
equation, suitable for numerical simulations. In the relativistic realm
such a Langevin process reads

Misplaced &

(1)

Here E=√m2+p2, and Γ is the drag or friction
coefficient. The covariance matrix, Cjk, of the fluctuating force
is related with the diffusion coefficients. Both coefficients are
dependent on (t,x,p) and are defined in the (local) rest
frame of the fluid. The ρk are Gaussian-normal distributed random
variables, i.e., its distribution function reads

P(ρ)=(12π)3/2exp(−ρ22).

(2)

The fluctuating force thus obeys

⟨F(fl)j(t)⟩=0,⟨F(fl)j(t)F(fl)k(t′)⟩=CjlCklδ(t−t′).

(3)

It is important to note that with these specifications the random
process is not yet uniquely determined since one has to specify, at
which argument of the momentum the covariance matrix Cjk has to be
taken to define the stochastic time integral in (1). Thus, we
set

Cjk=Cjk(t,x,p+ξdp).

(4)

For ξ=0, ξ=1/2, and ξ=1 the corresponding Langevin processes
are called the pre-point Ito, the mid-point Stratonovic-Fisk, and the
post-point Ito (or Hänggi-Klimontovich) realization.

The drag and diffusion coefficients for the heavy-quark propagation
within this framework are taken from a resonance approach
van Hees and Rapp (2005).

The initial production of charm quarks in our approach is based on a
Glauber approach. For the realization of the initial collision dynamics
we use the UrQMD model. We perform a first UrQMD run excluding
interactions between the colliding nuclei and save the nucleon-nucleon
collision space-time coordinates. These coordinates are used in a
second, full UrQMD run as possible production coordinates for the charm
quarks.

The momentum distribution for the initially produced charm quarks serves
as the starting point of our calculations. The pT distribution is
obtained from a fit to PYTHIA calculations. The fitting function for
charm quarks with 2.76TeV is:

dNd2pT=A11(1+A1⋅(p2T)2)A3

(5)

with the coefficients A1=0.136, A2=2.055 and A3=2.862.
Starting with this distribution as initial condition, at each
UrQMD/hydro time-step we perform an Ito-postpoint time-step, as
described in Sec. I. We use the UrQMD/hydro’s cell
velocities, cell temperature, the size of the time-step, and the
γ-factor for the calculation of the momentum transfer,
propagating all quarks independently. Our approach provides us only with
the charm-quark distribution. Since charm quarks cannot be measured
directly in experiments we include a hadronization mechanism for
D-Mesons, via the use of a quark-coalescence mechanism. To implement
this coalescence we perform our Langevin calculation until the
decoupling temperature is reached. Subsequently we add the momenta of
light quarks to those of the charm quarks. On average the velocity of
light quarks can be approximated by the flow-velocity vector of the
local hydro cell. The mass of the light quarks is assumed to be 369MeV so that the D-Meson mass becomes 1.869GeV when adding the
masses of the light quarks and the charm quarks (1.5GeV).

Figure 1: Flow v2 of D-Mesons in Pb+Pb collisions at √sNN=2.76TeV compared to data from the ALICE experiment
Bianchin (2011). A rapidity cut of |y|<0.35 is employed.

The D-Meson v2 exhibits a strong increase and reaches a maximum at
about pT=3GeV with v2∼15%. Considering the error bars the
agreement between the measurements and our calculation is quite
satisfactory.

A complementary view on the drag and diffusion coefficients is provided
by the nuclear suppression factor RAA. Figure 2 shows
the calculated nuclear modification factor RAA of D-Mesons at
LHC. Here we compare to two data sets available, for D0 and D+
mesons. In line with the experimental data the simulation is done for a
more central bin of σ/σto=0%-20%.

Figure 2: RAA of D-Mesons in Pb Pb collisions at √sNN=2.76TeV compared to preliminary data from ALICE
Rossi (2011). A rapidity cut of |y|<0.35 is employed.

We find a maximum of the RAA at about 2GeV followed by a
sharp decline to an RAA of about 0.2 at high pT. Especially at
low pT new measurements would be helpful to conduct a more detailed
comparison to the model prediction.

In summary we can conclude that our description of the medium
modification of charm quarks at LHC energies for both the elliptic flow
v2 and the nuclear modification factor RAA is compatible with
the experimental measurements of ALICE.

We are grateful to the Center for Scientific Computing (CSC) at
Frankfurt for providing computing resources. T. Lang gratefully
acknowledges support from the Helmholtz Research School on Quark Matter
Studies. This work is supported by the Hessian LOEWE initiative through
the Helmholtz International Center for FAIR (HIC for
FAIR). J. S. acknowledges a Feodor Lynen fellowship of the Alexander von
Humboldt foundation. This work is supported by the Office of Nuclear
Physics in the US Department of Energy’s Office of Science under
Contract No. DE-AC02-05CH11231. The computational resources have been
provided by the LOEWE Frankfurt Center for Scientific Computing
(LOEWE-CSC).